1. Field of the Invention
This invention relates to apparatuses, methods and systems for use in reservoir simulation. In particular, the invention provides methods, apparatuses and systems for more effectively and efficiently simulating fluid flow in reservoirs using an algebraic cascading class linear solver.
2. Background
Reservoir simulation often requires the numerical solution of the equations that describe the physics governing the complex behaviors of multi-component, multiphase fluid flow in natural porous media in the reservoir and other types of fluid flow elsewhere in the production system. The governing equations typically used to describe the fluid flow are based on the assumption of thermodynamic equilibrium and the principles of conservation of mass, momentum and energy, as described in Aziz & Settari, 1977. The complexity of the physics that govern reservoir fluid flow leads to systems of coupled nonlinear partial differential equations that are not amenable to conventional analytical methods. As a result, numerical solution techniques are necessary.
A variety of mathematical models, formulations, discretization methods, and solution strategies have been developed and are associated with a grid imposed upon an area of interest in a reservoir. Detailed discussions of the problems of reservoir simulation and the equations dealing with such problems can be found, for example, in a PCT published patent application to ExxonMobil, International Publication No. WO 01/40937, incorporated herein by reference and in U.S. Pat. No. 6,662,146 B1, incorporated herein by reference. If a grid imposed upon an area of interest in a reservoir is used, the grid may be structured or unstructured. Such grids are comprised of cells, each cell having one or more unknown properties, but with all the cells in the grid having pressure as an unknown. Other unknown properties include, but are not limited to, fluid properties such as water saturation or temperature, for example, or to “rock properties,” such as permeability or porosity to name a few. A cell treated as if it has only a single unknown (typically pressure) is called herein a “single variable cell,” while a cell with more than one unknown is called herein a “multi-variable cell.”
A matrix can be constructed to represent the gridded region of interest based on the different types of cells.
The following equation is used to solve for the unknown variables, called x:Ax=b  (Eq 1)where x is a block vector of the variables representing unknown properties of the cells and b is a block vector of known quantities. Block vector x and block vector b are the same length. Approaches to solving this problem include: Domain Multisector Decomposition (Cleve Ashcraft), Wirebasket Domain Decomposition (Barry Smith), Heirarchical interface decomposition (Henon & Saad), and GMRES (Saad & Shultz). But the problem specifically addressed by the current invention includes solving linear systems associated with large-scale heterogeneous problems for both structured and unstructured grids. The solution must be robust, computationally efficient with excellent convergence rate. The focus is on developing a scalable (i.e., efficient for very large problems) multi-level algebraic approach that relies completely on information contained in the linear system of equations that must be solved. Moreover, it is also essential that the solution be well suited for parallel computation. These requirements pose a great challenge, especially for strongly heterogeneous unstructured-grid problems, and existing methods, including Incomplete Upper-Lower (ILU) factorization and nested factorization, fall well short of meeting all of these essential requirements.